Finite Group Databasebeta

← #17017191

Subgroups — #17017191

Order 4352.

682 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14539243C211yesmaximal, characteristic
2#14538815C211yesmaximal, characteristic
3C2×#11763094C211yesmaximal, characteristic, complemented
4#14528278C212yesmaximal, complemented
5#14528256C212yesmaximal, complemented
6#14528154C212yesmaximal, complemented
7#14528132C212yesmaximal, complemented
8#14527536C211yesmaximal, characteristic, complemented
9#14527462C211yesmaximal, characteristic
10#14527457C211yesmaximal, characteristic, complemented
11Q8×M16×C17C211yesmaximal, characteristic, complemented
12C2×#10587595—21no—
13C2×#10587595—21no—
14#11763359C2211yescharacteristic
15C2×#10587555C2211yescharacteristic
16#11763203—22no—
17#11763189C2212yes—
18#11763182C2212yes—
19#11763127—21no—
20#11763094—21no—
21#11763094—21no—
22#11763094C2212yes—
23#11763094C2211yescharacteristic
24#11763094C2212yes—
25#11763094C2211yescharacteristic
26#11763009C2211yescharacteristic
27#11763009C2211yescharacteristic
28#11762948—21no—
29C4×#63230—21no—
30C4×#63230—21no—
31#11762862C2211yescharacteristic
32C2×#10587479—21no—
33C2×C4⋊#3622C2211yescharacteristic
34C2×#10587506—21no—
35#11762562C2212yes—
36#11762560C2212yes—
37#11762556C2212yes—
38#11762556C2212yes—
39#11762556C2212yes—
40#11762552C2212yes—
41C2×C4×Q8×C17C2211yescharacteristic
42C17×#446—22no—
43C17×#446C2212yes—
44C8×Q8×C17C2212yes—
45C17×#423—21no—
46C17×#423C2211yescharacteristic
47C4×M16×C17—21no—
48C4×M16×C17C2211yescharacteristic
49C22×Q8×C17D411yescharacteristic
50C2×#63252—21no—
51C22×Dic34—21no—
52C2×M16×C17—21no—
53C2×M16×C17D411yescharacteristic
54C2×M16×C17D411yescharacteristic
55C4×Q8×C17—21no—
56C4×Q8×C17—21no—
57C4×Q8×C17D411yescharacteristic
58C4×Q8×C17D411yescharacteristic
59C4×Q8×C17C2311yescharacteristic
60C4×Q8×C17C2311yescharacteristic
61C2×C4⋊C4×C17—21no—
62C2×C4⋊C4×C17D411yescharacteristic
63C2×C42×C17—21no—
64C2×C42×C17C2311yescharacteristic
65C2×#63230—21no—
66C2×#63230—21no—
67#10587616—21no—
68#10587616—21no—
69C2×C4⋊Dic17—21no—
70C2×C4⋊Dic17—21no—
71C22×#3622—21no—
72#10587595—44no—
73#10587595—44no—
74C2×Dic68—22no—
75C2×Dic68—22no—
76C2×#63219—22no—
77C2×#63219—22no—
78#10587561—21no—
79#10587560C2311yescharacteristic
80C4×D68—21no—
81C4×D68—21no—
82#10587555—21no—
83#10587555—21no—
84#10587555—21no—
85#10587555C2312yes—
86#10587555C2311yescharacteristic
87C17×C4⋊C8—22no—
88C17×C4⋊C8—22no—
89C17×C4⋊C8—21no—
90C17×C4⋊C8C2312yes—
91C17×#97—22no—
92C17×#97D412yes—
93C4×C8×C17—22no—
94C4×C8×C17C2312yes—
95#10587506—21no—
96#10587506—21no—
97#10587506—22no—
98#10587506—22no—
99C22⋊#3622—21no—
100#10587480—21no—
101#10587480—21no—
102#10587479—22no—
103#10587479—22no—
104#10587479—21no—
105#10587479—21no—
106C4⋊#3622—21no—
107C4⋊#3622C2311yescharacteristic
108C4⋊#3622C2311yescharacteristic
109C4⋊#3622C2312yes—
110C4×#3622—21no—
111C2×Q8×C17—21no—
112C2×Q8×C17—22no—
113C2×Q8×C17—41no—
114C2×Q8×C17C2×D411yescharacteristic
115C2×Q8×C17C2×D411yescharacteristic
116C22×C4×C17—21no—
117C22×C4×C17—21no—
118C22×C4×C17C2×D411yescharacteristic
119C22×C4×C17C2×D411yescharacteristic
120C22×C4×C17#4111yescharacteristic
121C2×C22⋊D17—41no—
122C22×Dic17—41no—
123#63252—42no—
124#63252—42no—
125C2×D68—21no—
126C2×D68—21no—
127C2×D68—21no—
128C2×C4×D17—41no—
129C2×Dic34—22no—
130C2×Dic34—21no—
131C2×Dic34—42no—
132C2×Dic34—42no—
133C2×Dic34—41no—
134M16×C17—22no—
135M16×C17—42no—
136M16×C17SD1614yescomplemented
137C2×C8×C17—22no—
138C2×C8×C17C2×D412yes—
139C2×C8×C17C2×D412yes—
140C4⋊C4×C17—21no—
141C4⋊C4×C17—21no—
142C4⋊C4×C17—21no—
143C4⋊C4×C17—21no—
144C4⋊C4×C17—21no—
145C4⋊C4×C17C2×D411yescharacteristic
146C4⋊C4×C17C2×D411yescharacteristic
147C42×C17—21no—
148C42×C17—21no—
149C42×C17—21no—
150C42×C17—21no—
151C42×C17C2×D411yescharacteristic
152C42×C17C2×D411yescharacteristic
153C42×C17#4111yescharacteristic
154C42×C17C2411yescharacteristic
155#63230—42no—
156#63230—42no—
157#63227—41no—
158#63227—41no—
159C4⋊Dic17—22no—
160C4⋊Dic17—21no—
161C4⋊Dic17—22no—
162C4⋊Dic17—21no—
163C4⋊Dic17—41no—
164C4⋊Dic17—41no—
165C2×#3622—21no—
166C2×#3622—21no—
167C2×#3622—22no—
168C2×#3622—42no—
169Dic68—44no—
170Dic68—44no—
171#63219—44no—
172#63219—44no—
173#19171—171nomaximal, Sylow
174C23×C17#12111yescharacteristic
175C22×D17—41no—
176Q8×C17—21no—
177Q8×C17—21no—
178Q8×C17—21no—
179Q8×C17—42no—
180Q8×C17#13712yescomplemented
181C2×C4×C17—21no—
182C2×C4×C17—21no—
183C2×C4×C17—21no—
184C2×C4×C17—21no—
185C2×C4×C17—21no—
186C2×C4×C17—21no—
187C2×C4×C17—41no—
188C2×C4×C17—42no—
189C2×C4×C17—81no—
190C2×C4×C17#12111yescharacteristic
191C2×C4×C17#12111yescharacteristic
192C2×C4×C17#12111yescharacteristic
193C2×C4×C17C2×SD1612yes—
194C2×C4×C17AGL(1,ℤ/8)12yes—
195C2×C4×C17C22×D411yescharacteristic
196C2×C4×C17C22×D411yescharacteristic
197C2×C4×C17C2×#4111yescharacteristic
198C22⋊D17—82no—
199C2×Dic17—42no—
200C2×Dic17—41no—
201C2×Dic17—82no—
202D68—41no—
203D68—41no—
204D68—41no—
205D68—41no—
206D68—81no—
207C4×D17—82no—
208Dic34—42no—
209Dic34—41no—
210Dic34—42no—
211Dic34—41no—
212Dic34—82no—
213C8×C17—22no—
214C8×C17—42no—
215C8×C17C2×SD1614yes—
216#3622—41no—
217#3622—42no—
218#3622—41no—
219#3152—171no—
220#3103—171no—
221C2×#475—171no—
222Q8×M16D1711yescharacteristic, complemented
223#1690—172no—
224#1686—172no—
225#1670—172no—
226#1666—172no—
227#1565—171no—
228#1547—171no—
229#1546—171no—
230C22×C17—21no—
231C22×C17—21no—
232C22×C17#46012yes—
233C22×C17C2×#12111yescharacteristic
234C2×D17—81no—
235C2×D17—81no—
236C2×D17—81no—
237C4×C17—21no—
238C4×C17—22no—
239C4×C17—21no—
240C4×C17—21no—
241C4×C17—41no—
242C4×C17—41no—
243C4×C17—41no—
244C4×C17#46012yes—
245C4×C17C22×SD1611yescharacteristic
246C4×C17C2×AGL(1,ℤ/8)11yescharacteristic
247C4×C17C2×#13711yescharacteristic
248C4×C17#57511yescharacteristic
249Dic17—82no—
250Dic17—81no—
251C2×#137—341no—
252C2×#137—341no—
253#532—171no—
254C2×C4⋊Q8—171no—
255C2×C4×Q8C2×D1711yescharacteristic
256#497—342no—
257#494—172no—
258#492—172no—
259#481—341no—
260#475—172no—
261#475—172no—
262#475—171no—
263#475—171no—
264#475—341no—
265#475—341no—
266#460—171no—
267#460—171no—
268#448—341no—
269#446—22no—
270#446C2×D1712yes—
271C8×Q8C2×D1712yes—
272C4×SD16—341no—
273C4×SD16—341no—
274#431—171no—
275C2×#106—341no—
276#423—21no—
277#423C2×D1711yescharacteristic
278C2×C4⋊C8—171no—
279C2×#103—341no—
280C4×M16—21no—
281C4×M16C2×D1711yescharacteristic
282#335—172no—
283#334—172no—
284#332—172no—
285#332—172no—
286#332—172no—
287#331—172no—
288C2×C17—22no—
289C2×C17C2×#46011yescharacteristic
290C2×C17#305611yescharacteristic
291C2×C17#306911yescharacteristic
292D17—161no—
293C2×#41—341no—
294C22×Q8—341no—
295C22×Q8D6811yescharacteristic
296#137—684no—
297#137—684no—
298C2×Q16—342no—
299C2×Q16—342no—
300C2×SD16—341no—
301C2×SD16—341no—
302C2×SD16—342no—
303C2×SD16—342no—
304C2×M16—21no—
305C2×M16C22⋊D1711yescharacteristic
306C2×M16C22⋊D1711yescharacteristic
307C22×C8—341no—
308C4⋊Q8—172no—
309C4⋊Q8—171no—
310C4⋊Q8—341no—
311C4⋊Q8—341no—
312C4⋊Q8—341no—
313#127—171no—
314#124—341no—
315#121—341no—
316#121—341no—
317C4×Q8—21no—
318C4×Q8—21no—
319C4×Q8D6811yescharacteristic
320C4×Q8D6811yescharacteristic
321C4×Q8C22×D1711yescharacteristic
322C4×Q8C22×D1711yescharacteristic
323C4×D4—341no—
324C4×D4—341no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—341no—
327C2×C4⋊C4—341no—
328C2×C4⋊C4D6811yescharacteristic
329C2×C42—21no—
330C2×C42C22×D1711yescharacteristic
331#106—342no—
332#106—342no—
333#106—341no—
334#106—341no—
335C4⋊C8—22no—
336C4⋊C8—22no—
337C4⋊C8—21no—
338C4⋊C8—172no—
339C4⋊C8—171no—
340C4⋊C8—171no—
341C4⋊C8—341no—
342C4⋊C8C22×D1712yes—
343#103—342no—
344#103—342no—
345#103—341no—
346#103—341no—
347#102—341no—
348#102—341no—
349C22⋊C8—341no—
350#97—22no—
351#97C22⋊D1712yes—
352C4×C8—22no—
353C4×C8—341no—
354C4×C8C22×D1712yes—
355C17#1917111yescharacteristic, complemented, Sylow
356#41—682no—
357#41—682no—
358C2×Q8—21no—
359C2×Q8—22no—
360C2×Q8—41no—
361C2×Q8—341no—
362C2×Q8—342no—
363C2×Q8—681no—
364C2×Q8—682no—
365C2×Q8—682no—
366C2×Q8C2×D6811yescharacteristic
367C2×Q8C2×D6811yescharacteristic
368C2×D4—341no—
369C2×D4—341no—
370C2×D4—341no—
371C2×D4—681no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—681no—
375C22×C4—681no—
376C22×C4C2×D6811yescharacteristic
377C22×C4#6325211yescharacteristic
378C22×C4C2×C22⋊D1711yescharacteristic
379Q16—684no—
380Q16—684no—
381SD16—684no—
382SD16—682no—
383SD16—682no—
384SD16—684no—
385M16—22no—
386M16—42no—
387M16#6323014yescomplemented
388C2×C8—22no—
389C2×C8—341no—
390C2×C8—341no—
391C2×C8—342no—
392C2×C8—682no—
393C2×C8C2×C22⋊D1712yes—
394C2×C8C2×C22⋊D1712yes—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—342no—
401C4⋊C4—341no—
402C4⋊C4—342no—
403C4⋊C4—341no—
404C4⋊C4—681no—
405C4⋊C4—681no—
406C4⋊C4C2×D6811yescharacteristic
407C4⋊C4C2×D6811yescharacteristic
408C22⋊C4—681no—
409C22⋊C4—681no—
410C42—21no—
411C42—21no—
412C42—21no—
413C42—21no—
414C42C2×D6811yescharacteristic
415C42#6325211yescharacteristic
416C42C2×C22⋊D1711yescharacteristic
417C42C23×D1711yescharacteristic
418C23—681no—
419C23#1058761611yescharacteristic
420Q8—21no—
421Q8—21no—
422Q8—21no—
423Q8—42no—
424Q8—682no—
425Q8—681no—
426Q8—682no—
427Q8—681no—
428Q8—1362no—
429Q8#1058759512yescomplemented
430D4—681no—
431D4—681no—
432D4—681no—
433D4—681no—
434D4—1361no—
435D4—1362no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—41no—
443C2×C4—42no—
444C2×C4—81no—
445C2×C4—681no—
446C2×C4—682no—
447C2×C4—1362no—
448C2×C4—1362no—
449C2×C4#1058761611yescharacteristic
450C2×C4#1058761611yescharacteristic
451C2×C4#1058761611yescharacteristic
452C2×C4C2×#6323012yes—
453C2×C4#1058763512yes—
454C2×C4C22×D6811yescharacteristic
455C2×C4C2×#6325211yescharacteristic
456C2×C4C22×C22⋊D1711yescharacteristic
457C8—22no—
458C8—42no—
459C8—681no—
460C8—682no—
461C8—681no—
462C8C2×#6323014yes—
463C22—21no—
464C22—21no—
465C22—1361no—
466C22—1361no—
467C22—1361no—
468C22#1176300912yes—
469C22C2×#1058761611yescharacteristic
470C4—21no—
471C4—21no—
472C4—21no—
473C4—22no—
474C4—41no—
475C4—41no—
476C4—41no—
477C4—1361no—
478C4—1362no—
479C4#1176300912yes—
480C4C22×#6323011yescharacteristic
481C4C2×#1058763511yescharacteristic
482C4C2×#1058759511yescharacteristic
483C4#1176360911yescharacteristic
484C2—22no—
485C2—2721no—
486C2C2×#1176300911yescharacteristic
487C2#1453841411yescharacteristic
488C2#1453852111yescharacteristic
subgroupG/H
centerC22C2×#10587616
commutator, cosocleC42×C17C24
FrattiniC42C23×D17
FittingQ8×M16×C17C2
socleC22×C17C2×#121
prime radical#11763359C22
hypercenterQ8×M16D17
hypocenterC17#19171
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ Q8×M16

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. C2×#11763094C2semidirect
4. #14528278C2semidirect
5. #14528256C2semidirect
6. #14528154C2semidirect
7. #14528132C2semidirect
8. #14527536C2semidirect
10. #14527457C2semidirect
11. Q8×M16×C17C2semidirect
136. M16×C17SD16semidirect
180. Q8×C17#137semidirect
222. Q8×M16D17semidirect
355. C17#19171semidirect
387. M16#63230semidirect
429. Q8#10587595semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#145392432
#145388152
C2×#117630942
#145282782
#145282562
#145281542
#145281322
#145275362
#145274622
#145274572
Q8×M16×C172
#1917117